Let $\mathcal X = \mathbb R^n$ and $\nu \in \mathcal P(\mathcal X)$ be a non-atomic Borel probability measure over $\mathcal X$ with finite second moment. Consider the following convex set, $$ \mathcal S_\nu = \{\eta\in\mathcal P(\mathcal X\times\mathcal X), ~(\mathrm{pr}_1)_*\eta = \nu, ~(\mathrm{pr}_2)_*((x-y)\eta) = 0 \}, $$ having noted $\mathrm{pr}_1(x,y) = x$, $\mathrm{pr}_2(x,y) = y$, and abusively $(x-y)\eta$ is the distribution defined by $(x-y)\mathrm d\eta(x,y)$. Essentially, $\eta\in\mathcal S_\nu$ if and only if $(X,Y)$ distributed according to $\eta$ are such that $X \sim\nu$ and $\mathbb E[X\mid Y] = Y$ (the estimate $\hat X$ of $X$ knowing $Y$, is $Y$).
My question is, how to characterize its extreme points?
So far, I found that when $\eta$ is constructed from $(X,Y)$ with $Y$ function of $X$, it is extreme. I also found that the converse holds if $(\mathrm{pr}_2)_*\eta$ has discrete support. However, I also have an example of $\eta$ extreme where $Y$ cannot be written as a function of $X$.